Subgroups — C₂×C₇₉×#2108 #34145012

Order 20224.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#2108	C₂	1	8	yes	maximal, direct product factor
5	C₂×#346×C₇₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₇₉	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₇₉	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₇₉	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₇₉	C₄	1	8	yes	complemented
11	C₂×M₃₂×C₇₉	C₂²	1	4	yes	—
12	C₂²×C₁₆×C₇₉	C₄	1	8	yes	complemented
13	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₇₉	C₂²	1	4	yes	—
16	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	#405×C₇₉	C₄	1	4	yes	—
18	#405×C₇₉	C₂²	1	4	yes	—
19	C₂×#97×C₇₉	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
21	#346×C₇₉	C₂²	1	8	yes	complemented
22	C₄×C₁₆×C₇₉	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₇₉	C₈	1	8	yes	—
25	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
33	#117×C₇₉	C₈	1	4	yes	—
34	#117×C₇₉	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₇₉	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₇₉	C₈	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	M₃₂×C₇₉	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₇₉	C₂×C₄	1	8	yes	—
42	C₂×C₁₆×C₇₉	C₂×C₄	1	16	yes	complemented
43	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
44	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
45	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
46	#97×C₇₉	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₇₉	C₂³	1	4	yes	—
49	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
51	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
57	M₁₆×C₇₉	C₂×C₈	1	16	yes	—
58	C₂×C₈×C₇₉	C₄²	1	4	yes	—
59	C₂×C₈×C₇₉	C₄²	1	8	yes	—
60	C₂×C₈×C₇₉	C₂×C₈	1	8	yes	—
61	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₇₉	C₂×C₈	1	8	yes	—
70	C₂²⋊C₄×C₇₉	C₂×C₈	1	8	yes	—
71	C₄²×C₇₉	C₂×C₈	1	4	yes	—
72	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
73	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
74	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
75	C₁₆×C₇₉	C₂²×C₄	1	8	yes	—
76	C₂³×C₇₉	—	2	4	no	—
77	C₂³×C₇₉	C₄×C₈	1	4	yes	—
78	C₂³×C₇₉	C₂²×C₈	1	2	yes	—
79	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	2	8	no	—
81	C₂×C₄×C₇₉	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₇₉	C₄×C₈	1	4	yes	—
83	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₇₉	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₇₉	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₇₉	C₂×C₄²	1	2	yes	—
92	C₈×C₇₉	C₂×C₄²	1	4	yes	—
93	C₈×C₇₉	C₂²×C₈	1	8	yes	—
94	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₇₉	—	2	8	no	—
97	C₂²×C₇₉	—	2	4	no	—
98	C₂²×C₇₉	C₂×C₄×C₈	1	4	yes	—
99	C₂²×C₇₉	C₂×C₄×C₈	1	2	yes	—
100	C₂²×C₇₉	#504	1	4	yes	—
101	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₇₉	—	2	8	no	—
103	C₄×C₇₉	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₇₉	C₂×C₄×C₈	1	4	yes	—
105	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₇₉	C₂³×C₈	1	1	yes	characteristic
107	C₂×#2108	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₇₉	—	2	4	no	—
109	C₂×C₇₉	#2108	1	4	yes	direct product factor
110	C₂×C₇₉	C₂²×C₄×C₈	1	1	yes	characteristic
111	C₂×C₇₉	C₂×#504	1	2	yes	—
112	C₂²×M₃₂	C₂×C₇₉	1	1	yes	characteristic
113	C₂³×C₁₆	C₂×C₇₉	1	1	yes	characteristic
114	C₂×#405	C₂×C₇₉	1	1	yes	characteristic
115	#2108	C₂×C₇₉	1	8	yes	direct product factor
116	C₂×#346	C₂×C₇₉	1	2	yes	complemented
117	C₂×C₄×C₁₆	C₂×C₇₉	1	2	yes	complemented
118	C₇₉	C₂×#2108	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×M₁₆	C₄×C₇₉	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₇₉	1	1	yes	characteristic
121	C₂×#117	C₄×C₇₉	1	1	yes	characteristic
122	C₂×M₃₂	C₄×C₇₉	1	8	yes	complemented
123	C₂×M₃₂	C₂²×C₇₉	1	4	yes	—
124	C₂²×C₁₆	C₄×C₇₉	1	8	yes	complemented
125	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₇₉	1	2	yes	—
127	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₇₉	1	4	yes	—
129	#405	C₄×C₇₉	1	4	yes	—
130	#405	C₂²×C₇₉	1	4	yes	—
131	C₂×#97	C₄×C₇₉	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₇₉	1	2	yes	—
133	#346	C₂²×C₇₉	1	8	yes	complemented
134	C₄×C₁₆	C₂²×C₇₉	1	8	yes	complemented
135	C₂³×C₄	C₂×C₄×C₇₉	1	1	yes	characteristic
136	C₂×M₁₆	C₈×C₇₉	1	8	yes	—
137	C₂×M₁₆	C₂×C₄×C₇₉	1	4	yes	—
138	C₂²×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₇₉	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₇₉	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₇₉	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₇₉	1	2	yes	—
143	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₇₉	1	2	yes	—
145	#117	C₈×C₇₉	1	4	yes	—
146	#117	C₂×C₄×C₇₉	1	4	yes	—
147	C₂×C₄⋊C₄	C₈×C₇₉	1	2	yes	—
148	C₂×C₂²⋊C₄	C₈×C₇₉	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₇₉	1	2	yes	—
150	M₃₂	C₂×C₄×C₇₉	1	16	yes	complemented
151	C₂×C₁₆	C₂×C₄×C₇₉	1	4	yes	—
152	C₂×C₁₆	C₂×C₄×C₇₉	1	4	yes	—
153	C₂×C₁₆	C₂×C₄×C₇₉	1	8	yes	—
154	C₂×C₁₆	C₂×C₄×C₇₉	1	16	yes	complemented
155	C₂×C₁₆	C₂³×C₇₉	1	2	yes	—
156	C₂×C₁₆	C₂³×C₇₉	1	4	yes	—
157	C₂×C₁₆	C₂³×C₇₉	1	2	yes	—
158	#97	C₂×C₄×C₇₉	1	8	yes	—
159	C₄×C₈	C₂×C₄×C₇₉	1	4	yes	—
160	C₄×C₈	C₂³×C₇₉	1	4	yes	—
161	C₂⁴	C₂×C₈×C₇₉	1	1	yes	characteristic
162	C₂²×C₄	C₄²×C₇₉	1	4	yes	—
163	C₂²×C₄	C₂×C₈×C₇₉	1	4	yes	—
164	C₂²×C₄	C₂×C₈×C₇₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₈×C₇₉	1	2	yes	—
166	C₂²×C₄	C₂×C₈×C₇₉	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₇₉	1	2	yes	—
169	M₁₆	C₂×C₈×C₇₉	1	16	yes	—
170	C₂×C₈	C₄²×C₇₉	1	4	yes	—
171	C₂×C₈	C₄²×C₇₉	1	8	yes	—
172	C₂×C₈	C₂×C₈×C₇₉	1	8	yes	—
173	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₇₉	1	2	yes	—
175	C₂×C₈	C₂²×C₄×C₇₉	1	2	yes	—
176	C₂×C₈	C₂²×C₄×C₇₉	1	2	yes	—
177	C₂×C₈	C₂²×C₄×C₇₉	1	4	yes	—
178	C₂×C₈	C₂²×C₄×C₇₉	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₇₉	1	4	yes	—
180	C₂×C₈	C₂⁴×C₇₉	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₈×C₇₉	1	8	yes	—
182	C₂²⋊C₄	C₂×C₈×C₇₉	1	8	yes	—
183	C₄²	C₂×C₈×C₇₉	1	4	yes	—
184	C₄²	C₂²×C₄×C₇₉	1	4	yes	—
185	C₁₆	C₂²×C₄×C₇₉	1	4	yes	—
186	C₁₆	C₂²×C₄×C₇₉	1	4	yes	—
187	C₁₆	C₂²×C₄×C₇₉	1	8	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₄×C₈×C₇₉	1	4	yes	—
190	C₂³	C₂²×C₈×C₇₉	1	2	yes	—
191	C₂³	C₂²×C₈×C₇₉	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	C₄×C₈×C₇₉	1	8	yes	—
194	C₂×C₄	C₄×C₈×C₇₉	1	4	yes	—
195	C₂×C₄	C₂×C₄²×C₇₉	1	4	yes	—
196	C₂×C₄	C₂×C₄²×C₇₉	1	2	yes	—
197	C₂×C₄	C₂²×C₈×C₇₉	1	8	yes	—
198	C₂×C₄	C₂²×C₈×C₇₉	1	2	yes	—
199	C₂×C₄	C₂²×C₈×C₇₉	1	4	yes	—
200	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₈×C₇₉	1	2	yes	—
202	C₂×C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
203	C₈	C₂×C₄²×C₇₉	1	2	yes	—
204	C₈	C₂×C₄²×C₇₉	1	4	yes	—
205	C₈	C₂²×C₈×C₇₉	1	8	yes	—
206	C₈	C₂³×C₄×C₇₉	1	1	yes	characteristic
207	C₈	C₂³×C₄×C₇₉	1	1	yes	characteristic
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	C₂×C₄×C₈×C₇₉	1	4	yes	—
211	C₂²	C₂×C₄×C₈×C₇₉	1	2	yes	—
212	C₂²	#504×C₇₉	1	4	yes	—
213	C₂²	C₂³×C₈×C₇₉	1	1	yes	characteristic
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×C₈×C₇₉	1	2	yes	—
216	C₄	C₂×C₄×C₈×C₇₉	1	4	yes	—
217	C₄	C₂²×C₄²×C₇₉	1	1	yes	characteristic
218	C₄	C₂³×C₈×C₇₉	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	C₇₉×#2108	1	4	yes	direct product factor
221	C₂	C₂²×C₄×C₈×C₇₉	1	1	yes	characteristic
222	C₂	C₂×#504×C₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆×C₇₉	C₂²
commutator	C₂	C₂²×C₄×C₈×C₇₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₇₉
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₇₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₇₉×#2108	C₂	direct (N × H)
5. C₂×#346×C₇₉	C₂	semidirect
6. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
10. C₂×M₃₂×C₇₉	C₄	semidirect
12. C₂²×C₁₆×C₇₉	C₄	semidirect
21. #346×C₇₉	C₂²	semidirect
22. C₄×C₁₆×C₇₉	C₂²	semidirect
38. M₃₂×C₇₉	C₂×C₄	semidirect
42. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
107. C₂×#2108	C₇₉	direct (N × H)
109. C₂×C₇₉	#2108	direct (N × H)
115. #2108	C₂×C₇₉	direct (N × H)
116. C₂×#346	C₂×C₇₉	semidirect
117. C₂×C₄×C₁₆	C₂×C₇₉	semidirect
118. C₇₉	C₂×#2108	direct (N × H)
122. C₂×M₃₂	C₄×C₇₉	semidirect
124. C₂²×C₁₆	C₄×C₇₉	semidirect
133. #346	C₂²×C₇₉	semidirect
134. C₄×C₁₆	C₂²×C₇₉	semidirect
150. M₃₂	C₂×C₄×C₇₉	semidirect
154. C₂×C₁₆	C₂×C₄×C₇₉	semidirect
220. C₂	C₇₉×#2108	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×M₃₂×C₇₉	2
C₂³×C₁₆×C₇₉	2
C₂×#405×C₇₉	2
C₇₉×#2108	2
C₂×#346×C₇₉	2
C₂×C₄×C₁₆×C₇₉	2
C₂×#2108	79

Subgroups — C2×C79×#2108 #34145012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₉×#2108 #34145012